Group #22550445 — subgroups

Subgroups — #22550445

Order 8128.

334 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14523248	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₅₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16861206	C₂	1	2	yes	maximal, complemented
4	#16861197	C₂	1	2	yes	maximal, complemented
5	#16861184	C₂	1	4	yes	maximal, complemented
6	#16861179	C₂	1	4	yes	maximal, complemented
7	#117×C₁₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×D₁₂₇	C₂²	1	2	yes	—
9	#14523248	—	2	4	no	—
10	C₂×Dic₂₅₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₅₀₈	—	2	4	no	—
12	C₂×D₅₀₈	C₂²	1	2	yes	—
13	C₂×D₅₀₈	C₂²	1	2	yes	—
14	C₂×D₅₀₈	C₂²	1	1	yes	characteristic, complemented
15	C₂×D₅₀₈	C₂²	1	8	yes	complemented
16	C₂×C₂²⋊D₁₂₇	C₂²	1	2	yes	—
17	C₂×C₄×D₁₂₇	C₂²	1	2	yes	complemented
18	(C₂×D₁₂₇)⋊C₄	C₂²	1	8	yes	complemented
19	C₂²×C₄×C₁₂₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₄×C₁₂₇	C₂²	1	2	yes	complemented
21	C₂²⋊C₄×C₁₂₇	C₂²	1	2	yes	—
22	C₄²×C₁₂₇	C₂²	1	2	yes	complemented
23	C₂²×D₁₂₇	—	2	4	no	—
24	C₂²×D₁₂₇	—	4	4	no	—
25	C₂²×D₁₂₇	C₂³	1	4	yes	—
26	C₂²×D₁₂₇	C₂³	1	2	yes	—
27	Dic₂₅₄	—	2	2	no	—
28	D₅₀₈	—	2	4	no	—
29	D₅₀₈	—	2	4	no	—
30	D₅₀₈	—	2	2	no	—
31	D₅₀₈	—	4	8	no	—
32	C₂²⋊D₁₂₇	—	2	4	no	—
33	C₂×Dic₁₂₇	C₂³	1	2	yes	—
34	C₄×D₁₂₇	—	2	4	no	—
35	C₂³×C₁₂₇	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₁₂₇	D₄	1	4	yes	complemented
37	C₂×C₄×C₁₂₇	C₂³	1	4	yes	—
38	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₁₂₇	C₂³	1	1	yes	characteristic
40	C₂²×C₁₂₇	—	2	2	no	—
41	C₂²×C₁₂₇	C₂×D₄	1	2	yes	—
42	C₂²×C₁₂₇	C₂⁴	1	1	yes	characteristic
43	C₂×D₁₂₇	—	2	2	no	—
44	C₂×D₁₂₇	—	2	4	no	—
45	C₂×D₁₂₇	—	2	4	no	—
46	C₂×D₁₂₇	—	2	8	no	—
47	C₂×D₁₂₇	—	4	8	no	—
48	C₄×C₁₂₇	—	2	4	no	—
49	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
50	C₄×C₁₂₇	C₂×D₄	1	2	yes	—
51	Dic₁₂₇	—	2	2	no	—
52	C₂×C₁₂₇	—	2	2	no	—
53	C₂×C₁₂₇	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₁₂₇	H(5,𝔽₂)	1	2	yes	—
55	D₁₂₇	—	4	2	no	—
56	D₁₂₇	—	4	4	no	—
57	C₁₂₇	#535	1	1	yes	characteristic, complemented, Sylow
58	#535	—	127	1	no	maximal, Sylow
59	C₂×#41	—	127	1	no	—
60	C₂²×D₄	—	127	1	no	—
61	#127	—	127	2	no	—
62	#124	—	127	2	no	—
63	#121	—	127	4	no	—
64	#120	—	127	4	no	—
65	#117	D₁₂₇	1	1	yes	characteristic, complemented
66	C₂⁴	—	127	2	no	—
67	#41	—	254	4	no	—
68	C₂×Q₈	—	127	1	no	—
69	C₂×D₄	—	127	2	no	—
70	C₂×D₄	—	127	8	no	—
71	C₂×D₄	—	127	2	no	—
72	C₂×D₄	—	127	2	no	—
73	C₂×D₄	—	127	1	no	—
74	C₂×D₄	—	254	4	no	—
75	C₂²×C₄	—	127	2	no	—
76	C₂²×C₄	C₂×D₁₂₇	1	1	yes	characteristic
77	C₄⋊C₄	C₂×D₁₂₇	1	2	yes	complemented
78	C₂²⋊C₄	—	127	8	no	—
79	C₂²⋊C₄	C₂×D₁₂₇	1	2	yes	—
80	C₄²	C₂×D₁₂₇	1	2	yes	complemented
81	C₂³	—	127	4	no	—
82	C₂³	—	127	2	no	—
83	C₂³	—	508	4	no	—
84	C₂³	—	254	4	no	—
85	C₂³	C₂²×D₁₂₇	1	1	yes	characteristic
86	Q₈	—	254	2	no	—
87	D₄	—	508	8	no	—
88	D₄	—	254	4	no	—
89	D₄	—	254	4	no	—
90	D₄	—	254	4	no	—
91	D₄	—	254	2	no	—
92	C₂×C₄	—	127	2	no	—
93	C₂×C₄	—	254	4	no	—
94	C₂×C₄	D₅₀₈	1	4	yes	complemented
95	C₂×C₄	C₂²×D₁₂₇	1	4	yes	—
96	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
97	C₂×C₄	C₂²×D₁₂₇	1	1	yes	characteristic
98	C₂²	—	2	2	no	—
99	C₂²	—	508	8	no	—
100	C₂²	—	254	2	no	—
101	C₂²	—	254	4	no	—
102	C₂²	—	254	4	no	—
103	C₂²	—	254	8	no	—
104	C₂²	C₂×D₅₀₈	1	2	yes	—
105	C₂²	C₂³×D₁₂₇	1	1	yes	characteristic
106	C₄	—	2	4	no	—
107	C₄	—	254	2	no	—
108	C₄	C₂×D₅₀₈	1	2	yes	—
109	C₄	C₂×D₅₀₈	1	2	yes	—
110	C₂	—	2	2	no	—
111	C₂	—	508	4	no	—
112	C₂	—	508	2	no	—
113	C₂	C₂²×D₅₀₈	1	1	yes	characteristic
114	C₂	#16861255	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14523248

C₂

yes

maximal, characteristic, complemented

C₂²×D₅₀₈

C₂

yes

maximal, characteristic, complemented

#16861206

C₂

yes

maximal, complemented

#16861197

C₂

yes

maximal, complemented

#16861184

C₂

yes

maximal, complemented

#16861179

C₂

yes

maximal, complemented

#117×C₁₂₇

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₂₇

C₂²

yes

—

#14523248

—

C₂×Dic₂₅₄

C₂²

yes

characteristic, complemented

C₂×D₅₀₈

—

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

—

C₂×D₅₀₈

C₂²

yes

characteristic, complemented

C₂×D₅₀₈

C₂²

yes

complemented

C₂×C₂²⋊D₁₂₇

C₂²

yes

—

C₂×C₄×D₁₂₇

C₂²

yes

complemented

(C₂×D₁₂₇)⋊C₄

C₂²

yes

complemented

C₂²×C₄×C₁₂₇

C₂²

yes

characteristic

C₄⋊C₄×C₁₂₇

C₂²

yes

complemented

C₂²⋊C₄×C₁₂₇

C₂²

yes

—

C₄²×C₁₂₇

C₂²

yes

complemented

C₂²×D₁₂₇

—

C₂²×D₁₂₇

—

C₂²×D₁₂₇

C₂³

yes

—

C₂²×D₁₂₇

C₂³

yes

—

Dic₂₅₄

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

D₅₀₈

—

C₂²⋊D₁₂₇

—

C₂×Dic₁₂₇

C₂³

yes

—

C₄×D₁₂₇

—

C₂³×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

D₄

yes

complemented

C₂×C₄×C₁₂₇

C₂³

yes

—

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂×C₄×C₁₂₇

C₂³

yes

characteristic

C₂²×C₁₂₇

—

C₂²×C₁₂₇

C₂×D₄

yes

—

C₂²×C₁₂₇

C₂⁴

yes

characteristic

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₂×D₁₂₇

—

C₄×C₁₂₇

—

C₄×C₁₂₇

C₂×D₄

yes

—

C₄×C₁₂₇

C₂×D₄

yes

—

Dic₁₂₇

—

C₂×C₁₂₇

—

C₂×C₁₂₇

C₂²×D₄

yes

characteristic

C₂×C₁₂₇

H(5,𝔽₂)

yes

—

D₁₂₇

—

D₁₂₇

—

C₁₂₇

#535

yes

characteristic, complemented, Sylow

#535

—

127

maximal, Sylow

C₂×#41

—

127

—

C₂²×D₄

—

127

—

#127

—

127

—

#124

—

127

—

#121

—

127

—

#120

—

127

—

#117

D₁₂₇

yes

characteristic, complemented

C₂⁴

—

127

—

#41

—

254

—

C₂×Q₈

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

127

—

C₂×D₄

—

254

—

C₂²×C₄

—

127

—

C₂²×C₄

C₂×D₁₂₇

yes

characteristic

C₄⋊C₄

C₂×D₁₂₇

yes

complemented

C₂²⋊C₄

—

127

—

C₂²⋊C₄

C₂×D₁₂₇

yes

—

C₄²

C₂×D₁₂₇

yes

complemented

C₂³

—

127

—

C₂³

—

127

—

C₂³

—

508

—

C₂³

—

254

—

C₂³

C₂²×D₁₂₇

yes

characteristic

Q₈

—

254

—

D₄

—

508

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

D₄

—

254

—

C₂×C₄

—

127

—

C₂×C₄

—

254

—

C₂×C₄

D₅₀₈

yes

complemented

C₂×C₄

C₂²×D₁₂₇

yes

—

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂×C₄

C₂²×D₁₂₇

yes

characteristic

C₂²

—

C₂²

—

508

—

100

C₂²

—

254

—

101

C₂²

—

254

—

102

C₂²

—

254

—

103

C₂²

—

254

—

104

C₂²

C₂×D₅₀₈

yes

—

105

C₂²

C₂³×D₁₂₇

yes

characteristic

106

C₄

—

107

C₄

—

254

—

108

C₄

C₂×D₅₀₈

yes

—

109

C₄

C₂×D₅₀₈

yes

—

110

C₂

—

111

C₂

—

508

—

112

C₂

—

508

—

113

C₂

C₂²×D₅₀₈

yes

characteristic

114

C₂

#16861255

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₂₇
commutator, socle	C₂²×C₁₂₇	C₂⁴
Fitting	#117×C₁₂₇	C₂
hypercenter	#117	D₁₂₇
hypocenter	C₁₂₇	#535
LCS	G ▷ C₂²×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ #117

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₂₇

commutator, socle

C₂²×C₁₂₇

C₂⁴

Fitting

#117×C₁₂₇

C₂

D₁₂₇

C₁₂₇

G ▷ C₂²×C₁₂₇ ▷ C₁₂₇

C₁ ◁ C₂² ◁ #117

N	H ≅ G/N	type
1. C₂×#14523248	C₂	semidirect
2. C₂²×D₅₀₈	C₂	semidirect
3. #16861206	C₂	semidirect
4. #16861197	C₂	semidirect
5. #16861184	C₂	semidirect
6. #16861179	C₂	semidirect
7. #117×C₁₂₇	C₂	semidirect
10. C₂×Dic₂₅₄	C₂²	semidirect
14. C₂×D₅₀₈	C₂²	semidirect
15. C₂×D₅₀₈	C₂²	semidirect
17. C₂×C₄×D₁₂₇	C₂²	semidirect
18. (C₂×D₁₂₇)⋊C₄	C₂²	semidirect
20. C₄⋊C₄×C₁₂₇	C₂²	semidirect
22. C₄²×C₁₂₇	C₂²	semidirect
36. C₂×C₄×C₁₂₇	D₄	semidirect
57. C₁₂₇	#535	semidirect
65. #117	D₁₂₇	semidirect
77. C₄⋊C₄	C₂×D₁₂₇	semidirect
80. C₄²	C₂×D₁₂₇	semidirect
94. C₂×C₄	D₅₀₈	semidirect

H ≅ G/N

type

1. C₂×#14523248

C₂

semidirect

2. C₂²×D₅₀₈

C₂

semidirect

3. #16861206

C₂

semidirect

4. #16861197

C₂

semidirect

5. #16861184

C₂

semidirect

6. #16861179

C₂

semidirect

7. #117×C₁₂₇

C₂

semidirect

10. C₂×Dic₂₅₄

C₂²

semidirect

14. C₂×D₅₀₈

C₂²

semidirect

15. C₂×D₅₀₈

C₂²

semidirect

17. C₂×C₄×D₁₂₇

C₂²

semidirect

18. (C₂×D₁₂₇)⋊C₄

C₂²

semidirect

20. C₄⋊C₄×C₁₂₇

C₂²

semidirect

22. C₄²×C₁₂₇

C₂²

semidirect

36. C₂×C₄×C₁₂₇

D₄

semidirect

57. C₁₂₇

#535

semidirect

65. #117

D₁₂₇

semidirect

77. C₄⋊C₄

C₂×D₁₂₇

semidirect

80. C₄²

C₂×D₁₂₇

semidirect

94. C₂×C₄

D₅₀₈

semidirect

Subgroups — #22550445

Conjugacy classes of subgroups

Complements